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Data Analysis

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Title: Data Analysis


1
Data Analysis
Chapter 8


2
In this chapter, we focus on 3 parts
Chapter 6
Data Analysis
  • 1. Descriptive Analysis
  • 2. Two-way Analysis of Variance
  • 3. Forecasting


3
1. Descriptive Analysis
Chapter 6
Data Analysis
  • 1.1 Index Numbers
  • 1.2 Exponential Smoothing


4
1.1 Index Numbers
Chapter 6
Data Analysis
  • Index Number a number that measures the change
    in a variable over time relative to the value of
    the variable during a specific base period
  • Simple Index Number index based on the relative
    changes (over time) in the price or quantity of a
    single commodity

5
1.1 Index Numbers
Chapter 6
Data Analysis
  • Laspeyres and Paasche Indexes compared
  • The Laspeyres Index weights by the purchase
    quantities of the baseline period
  • The Paasche Index weights by the purchase
    quantities of the period the index value
    represents.
  • Laspeyres Index is most appropriate when baseline
    purchase quantities are reasonable approximations
    of purchases in subsequent periods.
  • Paasche Index is most appropriate when you want
    to compare current to baseline prices at current
    purchase levels

6
1.1 Index Numbers
Chapter 6
Data Analysis
  • Calculating a Laspeyres Index
  • Collect price info for the k price series (the
    basket) to be used, denoted as P1t, P2tPkt
  • Select a base period t0
  • Collect purchase quantity info for base period,
    denoted as Q1t0, Q2t0..Qkt0
  • Calculate weighted totals for each time period
    using the formula
  • Calculate the index using the formula

7
1.1 Index Numbers
Chapter 6
Data Analysis
  • Calculating a Paasche Index
  • Collect price info for the k price series to be
    used, denoted as P1t, P2tPkt
  • Select a base period t0
  • Collect purchase quantity info for every period,
    denoted as Q1t, Q2t..Qkt
  • Calculate the index for time t using the formula

8
1.2 Exponential Smoothing
Chapter 6
Data Analysis
  • Exponential smoothing is a type of weighted
    average that applies a weight w to past and
    current values of the time series. (Yi actual
    value)
  • Exponential smoothing constant (w) lies between 0
    and 1, and smoothed series Et is calculated as
  • How much influence
  • does the past have when w 0 and
    when w 1?

9
1.2 Exponential Smoothing
Chapter 6
Data Analysis
  • Selection of smoothing constant w is made by
    researcher.
  • Small values of w give less weight to current
    value, yield a smoother series
  • Large values of w give more weight to current
    value, yield a more variable series

10
2 Two-way Analysis of Variance
Chapter 6
Data Analysis
  • Two-way ANOVA is a type of study design with one
    numerical outcome variable and two categorical
    explanatory variables.
  • Example In a completely randomised design we
    may wish to compare outcome by age, gender or
    disease severity. Subjects are grouped by one
    such factor and then randomly assigned one
    treatment.
  • Technical term for such a group is block and the
    study design is also called randomised block
    design

11
2 Two-way Analysis of Variance
Chapter 6
Data Analysis
  • 2.1 Randomised Block Design
  • 2.2 Analysis in Two-way ANOVA 1
  • 2.3 Analysis of Two-way ANOVA by the regression
    method


12
2.1 Randomised Block Design
Chapter 6
Data Analysis
  • Blocks are formed on the basis of expected
    homogeneity of response in each block (or group).
  • The purpose is to reduce variation in response
    within each block (or group) due to biological
    differences between individual subjects on
    account of age, sex or severity of disease.

13
2.1 Randomised Block Design
Chapter 6
Data Analysis
  • Randomised block design is a more robust design
    than the simple randomised design.
  • The investigator can take into account
    simultaneously the effects of two factors on an
    outcome of interest.
  • Additionally, the investigator can test for
    interaction, if any, between the two factors.

14
Steps in Planning a Randomised Block Design
Chapter 6
Data Analysis
2.1 Randomised Block Design
  1. Subjects are randomly selected to constitute a
    random sample.
  2. Subjects likely to have similar response
    (homogeneity) are put together to form a block.
  3. To each member in a block intervention is
    assigned such that each subject receives one
    treatment.
  4. Comparisons of treatment outcomes are made within
    each block

15
2.2 Analysis in Two-way ANOVA - 1
Chapter 6
Data Analysis
  • The variance (total sum of squares) is first
    partitioned into WITHIN and BETWEEN sum of
    squares. Sum of Squares BETWEEN is next
    partitioned by intervention, blocking and
    interaction

SS TOTAL
SS BETWEEN
SS WITHIN
SS INTERVENTION
SS BLOCKING
SS INTERACTION
16
Chapter 6
Data Analysis
2.2 Analysis in Two-way ANOVA - 1
method. And an interaction between gender and
teaching method is being sought. Analysis of
Two-way ANOVA is demonstrated in the slides that
follow. The study is about a n experiment
involving a teaching method in which professional
actors were brought in to play the role of
patients in a medical school. The test scores of
male and female students who were taught either
by the conventional method of lectures, seminars
and tutorials and the role-play method were
recorded. The hypotheses being tested
are Role-play method is superior to conventional
way of teaching. Female students in general have
better test scores than male students. Role-play
method makes a better impact on students of a
particular gender. Thus, there are two factors
gender and teaching method. And an interaction
between teaching method and gender is being
sought.
17
Chapter 6
Data Analysis
2.2 Analysis in Two-way ANOVA - 2
  • Each Sum of Squares (SS) is divided by its degree
    of freedom (df) to get the Mean Sum of Squares
    (MS).
  • The F statistic is computed for each of the three
    ratios as
  • MS INTERVENTION MS WITHIN
  • MS BLOCK MS WITHIN
  • MS INTERVENTION MS WITHIN

18
2.2 Analysis of Two-way ANOVA - 3
Chapter 6
Data Analysis
  • Analysis of Variance for score
  • Source DF SS MS F
    P
  • sex 1 2839 2839 22.75
    0.000
  • Tchmthd 1 1782 1782 14.28
    0.001
  • Error 29 3619 125
  • Total 31 8240

19
2.2 Analysis of Two-way ANOVA - 4
Chapter 6
Data Analysis
  • Individual 95 CI
  • Sex Mean -------------------------
    -------------
  • 0 58.5
    (------------)
  • 1 39.6 (-------------)
  • -------------------------
    -------------
  • 40.0 48.0
    56.0 64.0
  • Individual 95 CI
  • Tchmthd Mean -------------------------
    -------------
  • 0 56.5
    (--------------)
  • 1 41.6 (---------------)
  • -------------------------
    -------------
  • 42.0 49.0
    56.0 63.0

20
2.2 Analysis of Tw0-way ANOVA - 5
Chapter 6
Data Analysis
Analysis of Variance for SCORE Source
DF SS MS F
P SEX 1 2839
2839 22.64 0.000 TCHMTHD 1
1782 1782 14.21 0.001 INTERACTN
1 108 108 0.86
0.361 Error 28 3511
125 Total 31 8240
Interaction is not significant P 0.361
21
2.2 Analysis of Two-way ANOVA - 6
Chapter 6
Data Analysis
Individual 95 CI SEX Mean
-------------------------------------- 0
58.5
(------------) 1 39.6
(-------------)
--------------------------------------
40.0 48.0 56.0
64.0 Individual 95
CI TCHMTHD Mean ----------------------
---------------- 0 56.5
(--------------) 1
41.6 (---------------)
--------------------------------------
42.0 49.0 56.0
63.0
22
2.3 Analysis of Two-way ANOVA by the regression
method (reference coding)
Chapter 6
Data Analysis
The regression equation is SCORE 65.9 - 18.8
SEX - 14.9 TCHMTHD Predictor Coef
SE Coef T P Constant
65.913 3.420 19.27 0.000 SEX
-18.838 3.950 -4.77
0.000 TCHMTHD -14.925 3.950
-3.78 0.001 S 11.17 R-Sq 56.1
R-Sq(adj) 53.1 Analysis of Variance Source
DF SS MS F
P Regression 2 4620.9
2310.4 18.51 0.000 Residual Error 29
3619.0 124.8 Total 31
8239.8
23
2.3 Analysis of Two-way ANOVA by the regression
method (effect coding)
Chapter 6
Data Analysis
The regression equation is SCORE 49.0 - 9.42
EFCT-Sex - 7.46 EFCT-Tchmthd - 1.84
Interaction Predictor Coef SE
Coef T P Constant 49.031
1.980 24.77 0.000 EFCT-Sex
-9.419 1.980 -4.76 0.000 EFCT-Tch
-7.463 1.980 -3.77
0.001 Interact -1.838 1.980
-0.93 0.361 S 11.20 R-Sq 57.4
R-Sq(adj) 52.8
24
Reference Coding and Effect Coding - 1
Chapter 6
Data Analysis
  • In both methods, for k explanatory variables k-1
    dummy variables are created.
  • In reference coding the value 1 is assigned to
    the group of interest and 0 to all others (e.g.
    Female 1 Male 0).
  • In effect coding the value -1 is assigned to
    control group 1 to the group of interest (e.g.
    new treatment), and 0 to all others (e.g. Female
    1 Male (control group) -1 Role Play 1
    conventional teaching (control) -1).

25
Reference Coding and Effect Coding - 2
Chapter 6
Data Analysis
  • In reference coding the ß coefficients of the
    regression equation provide estimates of the
    differences in means from the control (reference)
    group for various treatment groups.
  • In effect coding the ß coefficients provide the
    differences from the overall mean response for
    each treatment group.

26
Chapter 6
Data Analysis
  • 3 Marketing Forecasting

3.1 The concept of market forecast 3.2 The
theoretical bases of forecast 3.3 The
classification of forecast methods 3.4
Qualitative Forecast Methods 3.5 Quantitative
Forecast Methods
27
3.1 The concept of market forecast
Chapter 6
Data Analysis
  • Based on market surveys and by applying
    scientific methods, to estimate the development
    situation of objects-forecasted in a certain
    period in future in order to help managers to
    improve decisions-making qualify. The process is
    generally called as market forecast.
  • In this chapter, objects-forecasted mainly are
    need quantities of products, sometime may also be
    product prices, competitive situations,
    environmental factors, and so on.

28
3.2 The theoretical bases of forecast
Chapter 6
Data Analysis
  • (1)The continuity principle
  • ?It is also called as inertia principle. Because
    of existing inertia, any system doesn't change
    its basic characteristics in the short run.
  • Attention all time series analysis methods
    are based on this principle.

29
Chapter 6
Data Analysis
3.2 The theoretical bases of forecast
  • (2)The analogy principle
  • ?time analogy to make an inference in future
    from the past and the present. When two things
    and more things have characteristic similarity
    (structure, mode, property, and develop
    tendency), we can forecast the developing things
    and the ready-to-develop things by studying the
    developed or advanced things. Attention analogy
    is suitable to the homogeneous things, also to
    inhomogeneous things.

30
Chapter 6
Data Analysis
3.2 The theoretical bases of forecast
  • (2)The analogy principle
  • ?(continual to front page) sampling analogy to
    make an inference about the whole from the part.
    When the whole and the part have characteristic
    similarity, we can forecast the whole by studying
    the part.
  • Attention the similarity is the key point either
    between the things with difference in advance
    time, or between the whole and the part.

31
Chapter 6
Data Analysis
3.2 The theoretical bases of forecast
  • (3)The relevancy principle
  • ?the theory considers that there is relativity
    among things, especially between two relevance
    things or causal things. All statistical
    regression analysis methods are based on this
    principle.

32
3.3 The classification of forecast methods
Chapter 6
Data Analysis
  • Although there are many theoretical forecast
    methods, in general forecast can be classified as
    two types
  • qualitative forecast
  • quantitative forecast.

33
3.3.1 Qualitative forecast
Chapter 6
Data Analysis
  • Qualitative forecast emphasizes the development
    tendencies (maybe essential characteristics), and
    is suitable to cases which there are a fewer and
    lack of data, such as science and technology
    forecast, development forecast of infant
    industries, long-term forecast, and forecasting
    things with uncertainty, etc.

34
Chapter 6
Data Analysis
3.3.2 Quantitative forecast
  • Quantitative forecast emphasizes the quantitative
    relationships of developing things. Essentially
    it is a kind of methods based on quantitative
    trend extrapolation, and is suitable to cases
    which there are many data.

35
Chapter 6
Data Analysis
3.3.3 The comparison of two methods
  • Qualitative forecast might contribute to the
    analysis of the basic trends, development
    inflection point, and the essence of things.
    Quantitative forecast can draw us numeral
    development concepts, and bring us conveniences
    of applying forecast results. None of two methods
    should be our preference, otherwise we probably
    abuse forecast methods.

36
3.4 Qualitative Forecast Methods
Chapter 6
Data Analysis
  • Delphi method
  • Social investigation or consumer survey
  • Colligating sellers opinions
  • having an informal discussion of a team
  • Integration of experts forecasts
  • The method of subjective probabilities
  • above methods all belong to non-models.

37
3.5 Quantitative Forecast Methods
Chapter 6
Data Analysis
  • Exponential Smoothing
  • ?mathematical model
  • ?signs and meanings to explain every sign and
    its meaning
  • ?avalue ais greater, means that the more late
    sample observations, the more its influence on
    forecast results. Vice versa. Recommendation a
    2/(n1)

38
3.5 Quantitative Forecast Methods
Chapter 6
Data Analysis
  • Exponential Smoothing
  • ?mathematical model----horizontal trend
  • ? mathematical model----lineal trend

39
3.5 Quantitative Forecast Methods
Chapter 6
Data Analysis
  • Exponential Smoothing
  • ?mathematical model---- quadratic curve trend

40
3.5 Quantitative Forecast Methods
Chapter 6
Data Analysis
  • Exponential Smoothing
  • ?how to choose mathematical models according to
    the trend of sample observations on coordinate
    diagram.

41
3.5 Quantitative Forecast Methods
Chapter 6
Data Analysis
  • Exponential Smoothing
  • ?how to determine initial values of smoothing
    parameters in general, the first observation
    value instead of them.
  • ?superiorities of exponential smoothing the
    storage data only is a fewer and it is suitable
    to forecast in short run.
  • ?application cases reference to another teaching
    materials.

42
3.5 Quantitative Forecast Methods
Chapter 6
Data Analysis
  • The growth curve
  • ?mathematical model
  • Logistic curve
  • Gompertz curve

43
3.5 Quantitative Forecast Methods
Chapter 6
Data Analysis
  • The growth curve
  • ?mathematical processing of initial observations
  • For Logistic curve
  • 2. For Gompertz curve

The processed data of observations can be used
for calculation of parameters k, a and b. The
calculation formulas are as following
44
3.5 Quantitative Forecast Methods
Chapter 6
Data Analysis
  • Calculation of k, a, and b

Attention the processed data of observations
must be blacked into 3 groups, thus we can obtain
3 sum values
When the number of initial data is not integer
multiple of 3, we must add or cut down data of
initials.
45
Chapter 6
Data Analysis
3.5 Quantitative Forecast Methods
  • Linear regression
  • ?An independent variable and a dependent variable
    are chosen on the model, and the varied relation
    of y and x is linear. This model is widely
    applied in quantitative forecasts.
  • ?the standard model
  • yabx
  • to non-standard equation, it is must
    transferred as standard model.

46
3.5 Quantitative Forecast Methods
Chapter 6
Data Analysis
  • Linear regression
  • ?determination of the coefficient a and b
  • by means of method of minimum squares, let
    the variance minimization, and the calculation
    of is as following
  • and let derivatives of Q to a and b are equal
    to 0, then

47
3.5 Quantitative Forecast Methods
Chapter 6
Data Analysis
  • Linear regression

We can get a and b
48
3.5 Quantitative Forecast Methods
Chapter 6
Data Analysis
  • Linear regression
  • ?then the forecast model is

It is necessary to check if the model the built
model is of high quality, the checking methods
are 1. standards error analysis
49
3.5 Quantitative Forecast Methods
Chapter 6
Data Analysis
  • Linear regression
  • in general, the following is required

2. correlation coefficient and test of
significance. The calculation of correlation
coefficient is
50
3.5 Quantitative Forecast Methods
Chapter 6
Data Analysis
  • Linear regression
  • ?discussion of correlation coefficient R
  • ?when R0, means y doesn't have the correlation
    with x, the case is called 0- correlation, so the
    built model cant be applied to forecast.
  • ?when R1, means y has the direct correlation
    with x.
  • ?in general, R is required to meet Rgt0.7. when
    Rlt0.3, means the built model can not be applied.
    When 0.3ltRlt0.7, means the model is not good and
    worthless.

51
3.5 Quantitative Forecast Methods
Chapter 6
Data Analysis
  • Linear regression
  • ? The quality of regression model is also tested
    by significance.
  • if , the built model is good and
    worth to application. on the contrary, if
    , the built model is worthless.
  • is the critical value of
    correlation coefficient R. It is known by looking
    up the given table. Theais given level of
    significance such as 0.05. The (n-m) is the
    degree of freedom such as n-2, m is the number of
    variables.

52
3.5 Quantitative Forecast Methods
Chapter 6
Data Analysis
  • Linear regression
  • ?the application of model if the future value of
    x is known as x?, the interval value of forecast
    variable is

Here, s?is determined by the formula
53
Quantitative Forecast Methods
Chapter 6
Data Analysis
  • Linear regression
  • and is T-distribution with
    significance level aand freedom degree n-m-1,
    here n is the number of observations, m is the
    number of variables.
  • ?In addition, many non-linear equation can be
    transferred as linear regression. For example

54
3.5 Quantitative Forecast Methods
Chapter 6
Data Analysis
  • Linear regression

Then we can get the equation , the
same work is suitable to exponential function,
logarithm function, reciprocal function, etc.
Those functions are called as allowed linear
regression with single variable.
Application case to see another teaching
materials.
55
3.5 Quantitative Forecast Methods
Chapter 6
Data Analysis
  • analogy forecasting method a case of
    application
  • ?We can forecast an object variable by
    researching the relationship between the variable
    and an economic indicator (for example, per
    capita national income, NI, or gross national
    product, GNP)
  • ?The relationship between vehicle population and
    NI is given in page 78 of textbook.

56
3.5 Quantitative Forecast Methods
Chapter 6
Data Analysis
  • Elastic coefficient method
  • ?For example, we can get the average growth rate
    of vehicle sales quantity by observing selling in
    the past years, but the rate is only an image. If
    we analyzes growth rate of sales together with
    growth rate of an economic indicator, we can
    improve forecast quality. Detail case is given in
    textbook.

57
3.5 Quantitative Forecast Methods
Chapter 6
Data Analysis
  • Combination forecasting method
  • ?Concept of combination forecasting it is called
    as combination forecasting to get a final
    forecast conclusion based on colligating multi
    intermediate forecast results gained by adopting
    multi-models, or on same model adopting multi
    independent variables.
  • ?The core idea combination is benefit to clear
    up the chanciness of single mode or independent
    variable.

58
3.5 Quantitative Forecast Methods
Chapter 6
Data Analysis
  • talking about forecast experience
  • ?policy variables it is very difficult to
    forecast changes of policy, but we can strengthen
    monitoring of environmental factors, especially
    paying attention to the running condition of the
    national economy. Establishing the monitoring and
    early warning system of the national economy is
    very necessary.

59
3.5 Quantitative Forecast Methods
Chapter 6
Data Analysis
  • talking about forecast experience
  • ?predicting accuracy and goodness of fit in
    model.
  • ?simple model and complexity model
  • ?single predicting result and many results
  • ?reliability of forecast conclusions three pints
    are very important----reality initial data
    (authoritativeness), accuracy of mathematical
    models, and correctness of forecast procedures.

60
3.5 Quantitative Forecast Methods
Chapter 6
Data Analysis
  • talking about forecast experience
  • ?data processing, actual cases and researchers
    imagination.
  • to improve forecast, establishing information
    system is very important.
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